WASHINGTON — President Trump appears all but certain to be confronted in coming weeks with a wrenching decision about Russia: whether to veto new, bipartisan sanctions against Moscow, partly for election interference that Mr. Trump has said is a fiction created by Democrats.

The sanctions, which would make it impossible for Mr. Trump to act alone to lift existing economic penalties imposed by President Barack Obama after Russia’s annexation of Crimea, were approved late Monday by Republicans and Democrats on the Senate Foreign Relations Committee. They have been embraced by Republican leaders, though not by the White House.

The agreement reached on Monday means the new sanctions are set to land on Mr. Trump’s desk just as his administration is fending off investigations into possible collusion with Russian officials during the campaign. Both Republicans and Democrats say they doubt Mr. Trump can afford to veto the bill.

But there is no question that the agreement reached in the Senate undercuts one of his stated goals.

In an interview last year with The New York Times, Mr. Trump raised questions about whether it was in the United States’ interest to continue the sanctions on Russia. He argued at the time that the United States seemed more concerned about Russia’s military activity in Ukraine than neighboring nations or Europe were.

The investigations underway in the House and the Senate, and by a Justice Department special counsel, are pursuing questions of whether Mr. Trump’s former national security adviser, Michael T. Flynn, discussed with the Russian ambassador the possibility of reversing sanctions that Mr. Obama imposed in the last days of his administration. Those sanctions sought to punish Russia for hacking that targeted the Democratic National Committee and state voting databases.

The new sanctions would make lifting those penalties almost impossible. They would cement the Ukraine sanctions, and impose new economic restrictions on “corrupt Russian actors,” officials involved in human rights abuses, suppliers of weapons to the Syrian government and anyone who conducts “malicious cyberactivity on behalf of the Russian government.”

For President Vladimir V. Putin of Russia, the passage of the new sanctions suggests that the larger goal of last year’s hacking and information warfare against the United States may have failed. If his strategic goal was to get sanctions lifted, he may enter his first meeting with Mr. Trump, expected this summer, with more sanctions coming at a moment of growing domestic unrest in Russia.

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Secretary of State Rex W. Tillerson before a hearing of the Senate Foreign Relations Committee on Tuesday.CreditBrendan Smialowski/Agence France-Presse — Getty Images

While it was unclear whether the House would accept the Senate version of the plan, it seemed likely that the Russia sanctions would remain in a final bill, appended to sanctions against Iran. A spokeswoman for Speaker Paul D. Ryan, Republican of Wisconsin, said he needed to review the legislation, but noted that he had supported efforts to impose sanctions on Russia in the past.

But on Capitol Hill on Tuesday, Secretary of State Rex W. Tillerson expressed at least some reservations about the legislation, suggesting to lawmakers that new sanctions could undercut the administration’s efforts to collaborate with Russia when it is in the United States’ interest.

“We have some channels that are open where we’re starting to talk,” Mr. Tillerson said, citing efforts to engage diplomatically with Russia over its role in Syria. “And I think what I wouldn’t want to do is close the channels off with something new.”

Before the Senate pursued bipartisan sanctions against Russia, Mr. Tillerson had requested more time before new penalties were proposed, hoping to use the administration’s early months to improve a relationship he said had reached a low point.

Some top Republicans had been inclined to give the White House time on the issue. But patience has worn thin — a shift made clear by the deal reached on Monday.

On Tuesday, Mr. Tillerson strained to demonstrate much progress, even after his trip to Moscow several months ago and a visit to Washington by Russia’s foreign minister, Sergey V. Lavrov.

“Our relationship’s at an all-time low, and it’s been deteriorating further,” he told senators on the Foreign Relations Committee, where he had been called to testify about the State Department’s budget. “Our objective is to stabilize that.”

In Syria, Mr. Tillerson said, American efforts to engage with Russia are “progressing in a positive way, but it is far too early in the process to say whether they’re going to bear fruit.”

He suggested that the administration would prefer “the flexibility to turn that heat up” on Russia sanctions if attempts to recalibrate the relationship with Moscow — the Trump administration will not use “reset,” a word favored by Hillary Clinton when she was secretary of state — ultimately falter.

“They have done plenty already that they should be responding to,” Senator Robert Menendez, Democrat of New Jersey, shot back.

Democrats have been quick to cite Mr. Trump’s friendlier posture toward Russia as a critical rationale for entrusting more review powers to Congress.

Senator Chuck Schumer of New York, the minority leader, said that it was “particularly significant” that a bipartisan coalition was seeking to re-establish Congress “as the final arbiter of sanctions relief.”

“Particularly,” he added from the Senate floor, “considering that this administration has been too eager to put sanctions relief on the table.” Mr. Schumer said he hoped the president would sign the legislation, “even though it cedes the power to Congress.”

Senator Jeanne Shaheen, Democrat of New Hampshire and a member of the Foreign Relations Committee, expressed relief that the bipartisan plan would ensure “that current sanctions cannot be unilaterally unwound by this administration.”

Senator Mitch McConnell of Kentucky, the majority leader, called the agreement “a signal action,” framing Russia’s efforts to interfere in last year’s election as “the result of eight years of a failed foreign policy” under Mr. Obama.

But in acknowledging the effort at interference, Mr. McConnell was essentially accepting the findings of the intelligence agencies. Mr. Trump briefly acknowledged those findings at the end of December, but ever since he has questioned the quality of the intelligence. His aides say he views any effort to accuse the Russians of trying to manipulate the election as part of a campaign to delegitimize his victory.

Lawmakers from both parties had expressed frustration that sanctions legislation had not been taken up sooner, in large part because Senator Bob Corker, Republican of Tennessee and the chairman of the Foreign Relations Committee, said he would give the administration space to make good on its pledge to reshape the Russia relationship. On Tuesday, Mr. Corker said, “I really do think we’ve ended up with a very good piece of legislation.”

Senator John McCain, Republican of Arizona, said this week that the delay in acting on Russia sanctions had been “unacceptable.”

Senator Benjamin L. Cardin of Maryland, the top Democrat on the Foreign Relations Committee, predicted that Mr. Trump would face immense pressure not to stand in the way of bipartisan legislation punishing Russia.

“I’d be very, very surprised if the president vetoes this bill,” he said before the deal was announced. “And he’s surprised me in a lot of different things.”

A version of this article appears in print on , on Page A11 of the New York edition with the headline: Congress Will Prod Trump to Punish Russia for Election Meddling He Calls a Fiction. Order Reprints | Today’s Paper | Subscribe